The Falcon 9 heavy rocket has a thrust of about 5,340,000 pounds (23700 kN) with a LEO payload of 140,660 pounds (65.7 t). The Space Shuttle, on the other hand, had a thrust of 6,780,000 pounds (30000 kN), but could only lift 53,700 pounds (24.2 t) of payload to LEO. If the Space Shuttle had more thrust, why did it have a significantly smaller payload?
The Space Shuttle also had to take itself in to space, which was a substantial amount of it's "payload" The dry Space Shuttle weights about 82 tons. Granted that includes the engine, and the payload for Falcon 9 doesn't include the upper stage, but that is the most significant difference.
Bottom line, the Space Shuttle was capable of getting more mass in to orbit. That mass had to include the life support of the astronauts, heat shield, and other related items, and thus couldn't carry as much payload to orbit.
Cargo transport wasn't the only function of the space shuttle. The orbiter was also a crew transport for up to 7 astronauts, was able to return to the ground and was also mostly reusable. All that functionality required significant amount of mass which it had to hurl into orbit in addition to its actual payload.
The Falcon 9 Heavy, on the other hand, is just a cargo launcher. The second stage has no other job than to push the payload into orbit and then ends as space trash. No crew cabin, no life support, no heat shield, no wings, no parts intended to work more than once (making the second stage reusable was considered once, but SpaceX came to the conclusion that it would be more trouble than it would be worth).
This makes the payload-to-mass ratio of the stage much better than that of the Space Shuttle orbiter. While the Space Shuttle orbiters had a dry mass of 68.5 tons[src], a Falcon 9 second stage only has a dry mass of 4.7 tons[src]. And this mass reduction benefits all the stages below.
Thrust is only one characteristic of a rocket, but not the only one contributing to a rocket's ability to lift payload into orbit. You also need to consider the rocket's mass, how efficiently it's burning its fuel, and how much fuel it can carry along, for example.
If you have two rockets with the same thrust, the same engines, and the same amount of fuel, but one weighing more (dry) than the other, the former rocket cannot deliver as much payload mass into orbit as the lighter rocket.
As @PearsonArtPhoto already wrote, the Space Shuttle not only needed to lift its payload into orbit, it also needed to lift itself into orbit. The Falcon Heavy on the other hand has much less additional mass in the last stage it needs to lift (like the fairing and other structural parts).
To take off, a rocket needs to generate more thrust than its mass (the thrust-to-weight ratio must be larger than 1 to take off). That's why the Space Shuttle needed to have a higher thrust: so it can actually leave the launch pad upwards. The Space Shuttle had a total mass of about 2,000 tonnes with a thrust of about 30,250kN at sea-level while the Falcon Heavy "only" has about 1,400 tonnes with a thrust of about 22,800kN at sea-level. That works out to a thrust-to-weight ratio of about 1.5 for the Space Shuttle and about 1.6 for the Falcon Heavy.